Approximating pointwise products of Laplacian eigenfunctions

Abstract

We consider Laplacian eigenfunctions on a d-dimensional bounded domain M (or a d-dimensional compact manifold M) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions (eℓ)ℓ∈N. We study the subspace of all pointwise productsAn=span{ei(x)ej(x):1≤i,j≤n}⊆L2(M). Clearly, that vector space has dimension dim(An)=n(n+1)/2. We prove that products eiej of eigenfunctions are simple in a certain sense: for any ε>0, there exists a low-dimensional vector space Bn that almost contains all products. More precisely, denoting the orthogonal projection ΠBn:L2(M)→Bn, we have∀1≤i,j≤n‖eiej−ΠBn(eiej)‖L2≤ε and the size of the space dim(Bn) is relatively small: for every δ>0,dim(Bn)≲M,δε−δn1+δ. We obtain the same sort of bounds for products of arbitrary length, as well for approximation in H−1 norm. Pointwise products of eigenfunctions are low-rank. This has implications, among other things, for the validity of fast algorithms in electronic structure computations.